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Nima Anari

Publications and source records attributed to Nima Anari.

At least 19 recordsLinked to original sources
arXiv2026

Beyond the Bethe Approximation of the Permanent

The canonical Bethe approximation gives a deterministic approximation to the permanent of every nonnegative matrix within a factor of $(\sqrt{2})^n$. We improve the base of this exponential factor: for some absolute constant $c<\sqrt{2}$, there is a deterministic polynomial-time $c^n$-approximation for the permanent of every nonnegative matrix. This shows that the canonical Bethe guarantee is not a barrier for deterministic approximation of the permanent. The proof augments the Bethe lower bound with a new certificate tailored to matrices on which that lower bound loses nearly the full factor. The author supplied the high-level plan of attack, and the proof was developed in an interaction with ChatGPT 5.6 Sol Pro. The author subsequently verified the results. Codex assisted with proof checking, manuscript assembly, and typesetting.

cs.DS↗
arXiv2026

Parallel Sampling from the Ising $p$-Spin Model

We study the parallel complexity of sampling from the high-temperature Ising mixed $p$-spin Gibbs measure, a canonical instance of a mean-field spin glass on the hypercube $\{\pm 1\}^n$. We propose two different algorithms for this problem, corresponding to two different regimes of accuracy. Our first algorithm is a parallel implementation of a Markov chain known as block dynamics, combined with an approximate rejection sampling step that uses an Ising model in a novel way as a proposal distribution to approximate the quadratic interaction terms of the $p$-spin Hamiltonian. For any $\varepsilon > 0$, this algorithm runs in $n^{\tfrac{1}{3}}\operatorname{polylog}(\tfrac{n}{\varepsilon})$ parallel time with $\operatorname{poly}(n, \log(\tfrac{1}{\varepsilon}))$ work, and outputs a sample whose law is $\varepsilon$-close to the $p$-spin measure in total variation distance. Our second algorithm uses Picard iterations to parallelize the Algorithmic Stochastic Localization (ASL) process of El Alaoui, Montanari, and Sellke (2025), and for any $\varepsilon > \varepsilon_n$, takes $\operatorname{polylog}(\tfrac{n}{\varepsilon})$ parallel time and $\operatorname{poly}(\tfrac{n}{\varepsilon})$ work to produce a sample that is $\varepsilon$-close to the $p$-spin measure in the normalized 2-Wasserstein metric. Here, $\varepsilon_n > 0$ is a threshold that goes to $0$ as $n \to \infty$. Our result constitutes a doubly exponential improvement in the $\varepsilon$ dependence of the runtime and an exponential improvement in the $\varepsilon$ dependence of the total work when compared to na\"ive ASL, whose runtime scales as $\exp(\operatorname{poly}(\tfrac{1}{\varepsilon}))$.

cs.DS↗
arXiv2026

G{\aa}rding's Theorem for Posynomials

We extend G{\aa}rding's theorem to homogeneous posynomials: if a finite positive sum of monomials with arbitrary nonnegative real exponents is zero-free on a product of right half-planes, then its degree-normalized root is concave. Consequently, zero-freeness in a sector of aperture $\alpha\pi$ implies $\alpha$-fractional log-concavity. This sharpens generic mixing and domain-sparsification guarantees for fixed-size matchings and nonsymmetric determinantal point processes. The result was developed in an AI-assisted interaction initiated and checked by the author; Codex also assisted with assembling and typesetting the manuscript.

cs.DS↗
arXiv2026

On Rounding on the Hypersimplex

We study correlated rounding on the hypersimplex, the base polytope of the uniform matroid. For each point \(x\) in the hypersimplex, the goal is to sample a \(k\)-subset \(A(x)\) with marginals \(x\), while coupling the samples for all choices of \(x\) so that nearby inputs produce nearby sets. We give conditional constant-stretch results for the natural maximum-entropy sequential scheme, based on a conjectured spectral property of the covariance matrix of the maximum-entropy distribution over \(k\)-subsets; this conjecture has been extensively tested numerically. Under this property, the scheme samples the maximum-entropy \(k\)-subset distribution with prescribed marginals using a common random ordering and common uniform thresholds. For every \(x,y\in[0,1]^n\) with \(\sum_i x_i=\sum_i y_i=k\), it satisfies \[ \mathbb{E}\!\left[|A(x)\triangle A(y)|\right] \le 6\|x-y\|_1 . \] Thus, conditional on the spectral hypothesis, the previous \(O(\log k)\) bound for hypersimplex correlated rounding would improve to a constant and the open question raised by Naor, Raju, Shetty, Srinivasan, Valieva, and Wajc would have a positive answer. By adding dummy coordinates, the same conditional result gives stretch at most \(12\) for the at-most-\(k\) polytope.

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arXiv2026

Sampling Directed Eulerian Tours in $\widetilde O(m^{3/2})$ Time

We give a randomized algorithm that samples a nearly uniform Eulerian tour of a directed Eulerian multigraph with $m$ arcs in $\widetilde O(m^{3/2})$ time. The guarantee is worst-case, applies to arbitrary directed Eulerian multigraphs, and breaks the $mn$-type arborescence-sampling barrier on sparse graphs. The core case is a $2$-in/$2$-out graph. We introduce a new local Markov chain, the flip--repair walk: one step locally splits a tour into two circuits and then chooses uniformly among the local flips that repair the state to one tour. We prove that this walk mixes in nearly linear many steps and implement the walk using a dynamic chord data structure. A pointwise degree-reduction wrapper extends the sampler from this degree-two core to arbitrary degrees while preserving the $\widetilde O(m^{3/2})$ total running time. The high-level algorithmic plan, the switching-network reduction, and the dynamic data-structure argument were devised by the author. The author conjectured the mixing theorem underlying the analysis, and GPT 5.5 Pro Extended produced its linear-algebra proof. Codex assisted with manuscript assembly and typesetting.

cs.DS↗
arXiv2026

Optimal $e^{(\gamma+o(1))n}$-Approximation of the Permanent of Positive Semidefinite Matrices

We determine, up to lower-order terms in the exponent, the best possible deterministic polynomial-time approximation ratio for the permanent of a Hermitian positive semidefinite matrix. If $A\succeq 0$ has no zero diagonal entry, $d=\operatorname{rank}(A)$, $A=VV^\dagger$ with $V\in\mathbb{C}^{n\times d}$ full column rank, and $v_1,\ldots,v_n$ are the rows of $V$, define \[ \Phi(V)=\max_{X\succ 0} \left\{\sum_{i=1}^n \log(v_i^\dagger Xv_i)+\log\det X-\operatorname{tr} X+d\right\}, \qquad \widehat P(A)=e^{\Phi(V)}. \] We prove the exact sandwich \[ e^{-\gamma n}\widehat P(A)\le \operatorname{per}(A)\le \widehat P(A). \] Here $\gamma$ is the Euler--Mascheroni constant. Since the maximization is concave, this gives a deterministic polynomial-time $e^{(\gamma+\varepsilon)n}$-approximation for every $\varepsilon>0$. Combined with the previous $e^{(\gamma-\varepsilon)n}$-hardness of approximation for positive semidefinite permanents, this resolves the optimal exponential approximation ratio for deterministic polynomial-time algorithms as $e^{(\gamma+o(1))n}$, assuming $\mathrm{P}\ne\mathrm{NP}$. The proof is an entropy argument applied to the standard Wick integral formula for $\operatorname{per}(A)$; the loss is exactly $\gamma$ per factor because $\mathbb{E}[\log T]=-\gamma$ for $T\sim\operatorname{Exp}(1)$. The result was obtained through interactions with GPT 5.5 Pro Extended: the first author's interaction was one-shot, while the second author's was a separate multi-turn interaction with high-level guidance. Both authors verified the theorem and proof. Codex was used to assemble and typeset the manuscript.

cs.DS↗
arXiv2026

Fast Spanning Tree Sampling in Broadcast Congested Clique

We present the first polylogarithmic-round algorithm for sampling a random spanning tree in the (Broadcast) Congested Clique model. For any constant $c > 0$, our algorithm outputs a sample from a distribution whose total variation distance from the uniform spanning tree distribution is at most $O(n^{-c})$ in at most $c \cdot \log^{O(1)}(n)$ rounds. The exponent hidden in $\log^{O(1)}(n)$ is an absolute constant independent of $c$ and $n$. This is an exponential improvement over the previous best algorithm of Pemmaraju, Roy, and Sobel (PODC 2025) for the Congested Clique model.

cs.DS↗
arXiv2025

Parallel Sampling via Autospeculation

We present parallel algorithms to accelerate sampling via counting in two settings: any-order autoregressive models and denoising diffusion models. An any-order autoregressive model accesses a target distribution $\mu$ on $[q]^n$ through an oracle that provides conditional marginals, while a denoising diffusion model accesses a target distribution $\mu$ on $\mathbb{R}^n$ through an oracle that provides conditional means under Gaussian noise. Standard sequential sampling algorithms require $\widetilde{O}(n)$ time to produce a sample from $\mu$ in either setting. We show that, by issuing oracle calls in parallel, the expected sampling time can be reduced to $\widetilde{O}(n^{1/2})$. This improves the previous $\widetilde{O}(n^{2/3})$ bound for any-order autoregressive models and yields the first parallel speedup for diffusion models in the high-accuracy regime, under the relatively mild assumption that the support of $\mu$ is bounded. We introduce a novel technique to obtain our results: speculative rejection sampling. This technique leverages an auxiliary ``speculative'' distribution~$\nu$ that approximates~$\mu$ to accelerate sampling. Our technique is inspired by the well-studied ``speculative decoding'' techniques popular in large language models, but differs in key ways. Firstly, we use ``autospeculation,'' namely we build the speculation $\nu$ out of the same oracle that defines~$\mu$. In contrast, speculative decoding typically requires a separate, faster, but potentially less accurate ``draft'' model $\nu$. Secondly, the key differentiating factor in our technique is that we make and accept speculations at a ``sequence'' level rather than at the level of single (or a few) steps. This last fact is key to unlocking our parallel runtime of $\widetilde{O}(n^{1/2})$.

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arXiv2025

Diffusion Models are Secretly Exchangeable: Parallelizing DDPMs via Autospeculation

Denoising Diffusion Probabilistic Models (DDPMs) have emerged as powerful tools for generative modeling. However, their sequential computation requirements lead to significant inference-time bottlenecks. In this work, we utilize the connection between DDPMs and Stochastic Localization to prove that, under an appropriate reparametrization, the increments of DDPM satisfy an exchangeability property. This general insight enables near-black-box adaptation of various performance optimization techniques from autoregressive models to the diffusion setting. To demonstrate this, we introduce \emph{Autospeculative Decoding} (ASD), an extension of the widely used speculative decoding algorithm to DDPMs that does not require any auxiliary draft models. Our theoretical analysis shows that ASD achieves a $\tilde{O} (K^{\frac{1}{3}})$ parallel runtime speedup over the $K$ step sequential DDPM. We also demonstrate that a practical implementation of autospeculative decoding accelerates DDPM inference significantly in various domains.

cs.LG↗
arXiv2024

Parallel Sampling via Counting

We show how to use parallelization to speed up sampling from an arbitrary distribution $\mu$ on a product space $[q]^n$, given oracle access to counting queries: $\mathbb{P}_{X\sim \mu}[X_S=\sigma_S]$ for any $S\subseteq [n]$ and $\sigma_S \in [q]^S$. Our algorithm takes $O({n^{2/3}\cdot \operatorname{polylog}(n,q)})$ parallel time, to the best of our knowledge, the first sublinear in $n$ runtime for arbitrary distributions. Our results have implications for sampling in autoregressive models. Our algorithm directly works with an equivalent oracle that answers conditional marginal queries $\mathbb{P}_{X\sim \mu}[X_i=\sigma_i\;\vert\; X_S=\sigma_S]$, whose role is played by a trained neural network in autoregressive models. This suggests a roughly $n^{1/3}$-factor speedup is possible for sampling in any-order autoregressive models. We complement our positive result by showing a lower bound of $\widetilde{\Omega}(n^{1/3})$ for the runtime of any parallel sampling algorithm making at most $\operatorname{poly}(n)$ queries to the counting oracle, even for $q=2$.

cs.DS↗
arXiv2024

Trickle-Down in Localization Schemes and Applications

Trickle-down is a phenomenon in high-dimensional expanders with many important applications -- for example, it is a key ingredient in various constructions of high-dimensional expanders or the proof of rapid mixing for the basis exchange walk on matroids and in the analysis of log-concave polynomials. We formulate a generalized trickle-down equation in the abstract context of linear-tilt localization schemes. Building on this generalization, we improve the best-known results for several Markov chain mixing or sampling problems -- for example, we improve the threshold up to which Glauber dynamics is known to mix rapidly in the Sherrington-Kirkpatrick spin glass model. Other applications of our framework include improved mixing results for the Langevin dynamics in the $O(N)$ model, and near-linear time sampling algorithms for the antiferromagnetic and fixed-magnetization Ising models on expanders. For the latter application, we use a new dynamics inspired by polarization, a technique from the theory of stable polynomials.

math.PR↗
arXiv2024

Batch Active Learning of Reward Functions from Human Preferences

Data generation and labeling are often expensive in robot learning. Preference-based learning is a concept that enables reliable labeling by querying users with preference questions. Active querying methods are commonly employed in preference-based learning to generate more informative data at the expense of parallelization and computation time. In this paper, we develop a set of novel algorithms, batch active preference-based learning methods, that enable efficient learning of reward functions using as few data samples as possible while still having short query generation times and also retaining parallelizability. We introduce a method based on determinantal point processes (DPP) for active batch generation and several heuristic-based alternatives. Finally, we present our experimental results for a variety of robotics tasks in simulation. Our results suggest that our batch active learning algorithm requires only a few queries that are computed in a short amount of time. We showcase one of our algorithms in a study to learn human users' preferences.

cs.LG↗
arXiv2024

Fast parallel sampling under isoperimetry

We show how to sample in parallel from a distribution $\pi$ over $\mathbb R^d$ that satisfies a log-Sobolev inequality and has a smooth log-density, by parallelizing the Langevin (resp. underdamped Langevin) algorithms. We show that our algorithm outputs samples from a distribution $\hat\pi$ that is close to $\pi$ in Kullback--Leibler (KL) divergence (resp. total variation (TV) distance), while using only $\log(d)^{O(1)}$ parallel rounds and $\widetilde{O}(d)$ (resp. $\widetilde O(\sqrt d)$) gradient evaluations in total. This constitutes the first parallel sampling algorithms with TV distance guarantees. For our main application, we show how to combine the TV distance guarantees of our algorithms with prior works and obtain RNC sampling-to-counting reductions for families of discrete distribution on the hypercube $\{\pm 1\}^n$ that are closed under exponential tilts and have bounded covariance. Consequently, we obtain an RNC sampler for directed Eulerian tours and asymmetric determinantal point processes, resolving open questions raised in prior works.

cs.DS↗
arXiv2023

Universality of Spectral Independence with Applications to Fast Mixing in Spin Glasses

We study Glauber dynamics for sampling from discrete distributions $\mu$ on the hypercube $\{\pm 1\}^n$. Recently, techniques based on spectral independence have successfully yielded optimal $O(n)$ relaxation times for a host of different distributions $\mu$. We show that spectral independence is universal: a relaxation time of $O(n)$ implies spectral independence. We then study a notion of tractability for $\mu$, defined in terms of smoothness of the multilinear extension of its Hamiltonian -- $\log \mu$ -- over $[-1,+1]^n$. We show that Glauber dynamics has relaxation time $O(n)$ for such $\mu$, and using the universality of spectral independence, we conclude that these distributions are also fractionally log-concave and consequently satisfy modified log-Sobolev inequalities. We sharpen our estimates and obtain approximate tensorization of entropy and the optimal $\widetilde{O}(n)$ mixing time for random Hamiltonians, i.e. the classically studied mixed $p$-spin model at sufficiently high temperature. These results have significant downstream consequences for concentration of measure, statistical testing, and learning.

math.PR↗
arXiv2023

Parallel Sampling of Diffusion Models

Diffusion models are powerful generative models but suffer from slow sampling, often taking 1000 sequential denoising steps for one sample. As a result, considerable efforts have been directed toward reducing the number of denoising steps, but these methods hurt sample quality. Instead of reducing the number of denoising steps (trading quality for speed), in this paper we explore an orthogonal approach: can we run the denoising steps in parallel (trading compute for speed)? In spite of the sequential nature of the denoising steps, we show that surprisingly it is possible to parallelize sampling via Picard iterations, by guessing the solution of future denoising steps and iteratively refining until convergence. With this insight, we present ParaDiGMS, a novel method to accelerate the sampling of pretrained diffusion models by denoising multiple steps in parallel. ParaDiGMS is the first diffusion sampling method that enables trading compute for speed and is even compatible with existing fast sampling techniques such as DDIM and DPMSolver. Using ParaDiGMS, we improve sampling speed by 2-4x across a range of robotics and image generation models, giving state-of-the-art sampling speeds of 0.2s on 100-step DiffusionPolicy and 14.6s on 1000-step StableDiffusion-v2 with no measurable degradation of task reward, FID score, or CLIP score.

cs.LG↗
arXiv2023

Distortion in metric matching with ordinal preferences

Suppose that we have $n$ agents and $n$ items which lie in a shared metric space. We would like to match the agents to items such that the total distance from agents to their matched items is as small as possible. However, instead of having direct access to distances in the metric, we only have each agent's ranking of the items in order of distance. Given this limited information, what is the minimum possible worst-case approximation ratio (known as the distortion) that a matching mechanism can guarantee? Previous work by Caragiannis et al. proved that the (deterministic) Serial Dictatorship mechanism has distortion at most $2^n - 1$. We improve this by providing a simple deterministic mechanism that has distortion $O(n^2)$. We also provide the first nontrivial lower bound on this problem, showing that any matching mechanism (deterministic or randomized) must have worst-case distortion $\Omega(\log n)$. In addition to these new bounds, we show that a large class of truthful mechanisms derived from Deferred Acceptance all have worst-case distortion at least $2^n - 1$, and we find an intriguing connection between thin matchings (analogous to the well-known thin trees conjecture) and the distortion gap between deterministic and randomized mechanisms.

cs.GT↗
arXiv2022

Optimal Sublinear Sampling of Spanning Trees and Determinantal Point Processes via Average-Case Entropic Independence

We design fast algorithms for repeatedly sampling from strongly Rayleigh distributions, which include random spanning tree distributions and determinantal point processes. For a graph $G=(V, E)$, we show how to approximately sample uniformly random spanning trees from $G$ in $\widetilde{O}(\lvert V\rvert)$ time per sample after an initial $\widetilde{O}(\lvert E\rvert)$ time preprocessing. For a determinantal point process on subsets of size $k$ of a ground set of $n$ elements, we show how to approximately sample in $\widetilde{O}(k^\omega)$ time after an initial $\widetilde{O}(nk^{\omega-1})$ time preprocessing, where $\omega<2.372864$ is the matrix multiplication exponent. We even improve the state of the art for obtaining a single sample from determinantal point processes, from the prior runtime of $\widetilde{O}(\min\{nk^2, n^\omega\})$ to $\widetilde{O}(nk^{\omega-1})$. In our main technical result, we achieve the optimal limit on domain sparsification for strongly Rayleigh distributions. In domain sparsification, sampling from a distribution $\mu$ on $\binom{[n]}{k}$ is reduced to sampling from related distributions on $\binom{[t]}{k}$ for $t\ll n$. We show that for strongly Rayleigh distributions, we can can achieve the optimal $t=\widetilde{O}(k)$. Our reduction involves sampling from $\widetilde{O}(1)$ domain-sparsified distributions, all of which can be produced efficiently assuming convenient access to approximate overestimates for marginals of $\mu$. Having access to marginals is analogous to having access to the mean and covariance of a continuous distribution, or knowing "isotropy" for the distribution, the key assumption behind the Kannan-Lov\'asz-Simonovits (KLS) conjecture and optimal samplers based on it. We view our result as a moral analog of the KLS conjecture and its consequences for sampling, for discrete strongly Rayleigh measures.

cs.DS↗
arXiv2022

Quadratic Speedups in Parallel Sampling from Determinantal Distributions

We study the problem of parallelizing sampling from distributions related to determinants: symmetric, nonsymmetric, and partition-constrained determinantal point processes, as well as planar perfect matchings. For these distributions, the partition function, a.k.a. the count, can be obtained via matrix determinants, a highly parallelizable computation; Csanky proved it is in NC. However, parallel counting does not automatically translate to parallel sampling, as classic reductions between the two are inherently sequential. We show that a nearly quadratic parallel speedup over sequential sampling can be achieved for all the aforementioned distributions. If the distribution is supported on subsets of size $k$ of a ground set, we show how to approximately produce a sample in $\widetilde{O}(k^{\frac{1}{2} + c})$ time with polynomially many processors for any constant $c>0$. In the two special cases of symmetric determinantal point processes and planar perfect matchings, our bound improves to $\widetilde{O}(\sqrt k)$ and we show how to sample exactly in these cases. As our main technical contribution, we fully characterize the limits of batching for the steps of sampling-to-counting reductions. We observe that only $O(1)$ steps can be batched together if we strive for exact sampling, even in the case of nonsymmetric determinantal point processes. However, we show that for approximate sampling, $\widetilde{\Omega}(k^{\frac{1}{2}-c})$ steps can be batched together, for any entropically independent distribution, which includes all mentioned classes of determinantal point processes. Entropic independence and related notions have been the source of breakthroughs in Markov chain analysis in recent years, so we expect our framework to prove useful for distributions beyond those studied in this work.

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